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where
b
,
u
i
m
. Considering that the basis of the
m
unit vectors
u
i
spans the
∈
m
vector space but in this last expression only the
n
unit vectors
u
i
corre-
sponding to the nonnull singular values
entire
σ
i
are to be taken into account, it follows
that for
m
>
n
,
i
=
1
u
i
b
2
expresses the squared norm of the projection of the
vector
b
onto the subspace spanned by
n
unit vectors
u
i
. Hence, this squared norm
is less than the squared norm of
b
except when
m
=
n
, where this expression
represents the squared norm of
b
, so eq. (5.68) is valid with the sign equal.
5.3.5 Approximated Analysis of the Relative GeTLS Solution Positions
The GeTLS solutions comprise solutions of the GeTLS problems
∀
ζ
∈
[0, 1],in
particular the OLS, TLS, and DLS solutions. Here, the basic assumptions are:
1. An overdetermined system of only slightly incompatible equations (i.e.,
γ
1).
2. In the expression of
g
(γ
,
ζ )
,the
n
th term of
i
=
1
q
i
/λ
i
−
2
γζ
is domi-
nant (i.e.,
λ
n
and
λ
n
−
1
are enough distant) for
γ<λ
n
/
2
ζ
.
min
With these assumptions,
g
(γ
,
ζ )
becomes
q
n
λ
n
−
2
γζ
+
2
γ(
1
−
ζ)
−
b
T
b
g
(γ
,
ζ )
≈
(5.70)
Its zero for
γ<λ
n
/
2
ζ
gives the solution (minimum) for the corresponding
GeTLS problem. Hence, the GeTLS solution solves the approximated second-
degree equation
−
2
ζ
b
T
b
+
(
1
−
ζ ) λ
n
γ
+
λ
n
b
T
b
−
q
n
≈
0
2
4
ζ (
1
−
ζ ) γ
(5.71)
which, by using the first assumption, can be simplified and gives the approximated
solution:
n
b
T
b
q
n
1
2
λ
−
1
2
k
ζ
b
T
b
+
(
γ
min
≈
ζ
b
T
b
+
(
1
−
ζ ) λ
n
=
1
−
ζ ) λ
n
(5.72)
where
k
≥
0 for the property (5.68). The three main solutions are
$
k
γ
OLS
≈
(ζ
=
0
)
2
λ
n
k
b
T
b
+
λ
n
γ
TLS
≈
(ζ
=
.
)
γ
min
=
0
5
(5.73)
%
k
2
b
T
b
γ
DLS
≈
(ζ
=
1
)
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